The acoustical analog of the diffraction grating, which has played an important part in spectroscopy for over 100 years, was not used in architectural acoustics until the invention and development of the reflection-phase grating diffusor, within the past decade. The one-dimensional reflection-phase grating, described in U.S. Pat. No. D291,601 and shown in FIG. 1, consists of a linear periodic grouping of an array of wells of equal width, but different depths, separated by thin dividers. The depths of the wells are determined through calculations using the quadratic residue number theory. In a one-dimensional reflection-phase grating, the number theoretic phase variation occurs in one direction on the face of the unit and is invariant 90.degree. from that direction. The reflection-phase grating can also be designed in a two-dimensional realization where the number theoretic phase variation occurs in two orthogonal directions, as opposed to in only one. As in the case of the one-dimensional diffusor, quadratic-residue well depth sequences have been used. A two-dimensional diffusor consists of a two-dimensional array of square, rectangular or circular wells of varying depths, separated by thin dividers. FIG. 2 shows a two-dimensional quadratic-residue diffusor, marketed under the Registered Trademark "Omniffusor", which is described in U.S. Pat. No. D306,764. It can be seen that the "Omniffusor" diffusor possesses two vertical mirror planes of symmetry and four-fold rotational symmetry, while, as will be explained in detail hereinafter, the primitive root diffusor contains no symmetry elements.
A schematic comparison between the hemidisk coverage pattern of a one-dimensional quadratic-residue diffusor and the hemispherical coverage pattern of a two-dimensional quadratic-residue diffusor is shown in FIGS. 3 and 4, respectively. In FIG. 3, the incident plane wave is indicated with arrows arriving at 45.degree. with respect to the surface normal. The radiating arrows touching the hemidisk envelope indicate the diffraction directions. In FIG. 4, the incident plane wave is indicated with arrows arriving at 45.degree. with respect to the surface normal. The arrows radiating from the hemisphere envelope indicate a few of the many diffraction directions.
While the quadratic-residue sequences provide uniform diffusion in all of the diffraction orders, the primitive root sequence suppresses the zero order and the Zech logarithm suppress the zero and first diffraction orders, at the design frequency and integer multiples thereof. Applicants have found that the scattering intensity pattern for the primitive root sequence omits the specular lobe, which lobe is present in the scattering intensity pattern of a quadratic-residue number theory sequence. ##EQU1##
The diffraction directions for each wavelength, .lambda., of incident sound scattered from a reflection-phase grating (FIG. 5) are determined by the dimension of the repeat unit NW, Equation 1. N being the number of wells per period, W being the width of the well, .alpha..sub.i being the angle of incidence, .alpha..sub.d being the angle of diffraction, and n being the diffraction order. The intensity in any direction (FIG. 6) is determined by the Fourier transform of the reflection factor, r.sub.h, which is a function of the depth sequence (d.sub.h) or phases within a period (Equation 2). Equation 1 indicates that as the repeat unit NW increases, more diffraction lobes are experienced and the diffusion increases. In addition, as the number of periods increases, the energy is concentrated into the diffraction directions (FIG. 6).
FIG. 6(top) shows the theoretical scattering intensity pattern for a quadratic-residue diffusor. Diffraction directions are represented as dashed lines; scattering from finite diffusor occurs over broad lobes. Maximum intensity has been normalized to 50 dB. In FIG. 6(middle), the number of periods has been increased from 2 to 25, concentrating energy into diffraction directions. In FIG. 6(bottom), the number of wells per period has been increased from 17 to 89, thereby increasing number of lobes by a factor of 5. Arrows indicate incident and specular reflection directions.
The reflection-phase grating behaves like an ideal diffusor in that the surface irregularities provide excellent time distribution of the backscattered sound and uniform wide-angle coverage over a broad designable frequency bandwidth, independent of the angle of incidence. The diffusing properties are in effect invariant to the incident frequency, the angle of incidence and the angle of observation.
The well depths for the one-dimensional quadratic-residue diffusor, Equation 3, and the two-dimensional quadratic-residue diffusor, Equation 4, are based on mathematical number-theory sequences, which have the unique property that the Fourier transform of the exponentiated sequence values has constant magnitude in the diffraction directions. The symbol h represents the well number in the one-dimensional quadratic-residue diffusor and the symbols h and k represent the well number in the two-dimensional quadratic-residue diffusor
For the quadratic sequence elements, S.sub.h =h.sup.2.sub.modN and S.sub.h,k ={h.sup.2 +k.sup.2 }.sub.modN' where N is an odd prime. For example, if N=7, the one-dimensional sequence elements, for h=0-6 are 0,1,4,2,2,4,1. For higher values of h, the sequence repeats. Values of S.sub.h,k for N=7 are given in Table 1 for a two-dimensional quadratic-residue diffusor.
TABLE 1 ______________________________________ 0 1 4 2 2 4 1 1 2 5 3 3 5 2 4 5 1 6 6 1 5 2 3 6 4 4 6 3 2 3 6 4 6 3 2 4 5 1 6 6 1 5 1 2 5 3 3 5 2 ##STR1## (5) ______________________________________
The two-dimensional polar response or diffraction orders (m,n), Equation 5, can be conveniently displayed in a reciprocal lattice reflection phase grating plot, shown in FIG. 7. The diffraction orders are determined by the constructive interference condition.
When the depth variations are defined by a quadratic residue sequence, the non-evanescent scattering lobes are represented as equal energy contours within a circle whose radius is equal to the non-dimensional quantity, NW/.lambda.. This is a convenient plot because the effects of changing the frequency can easily be seen. Thus, if .lambda..sub.2 is decreased to .lambda..sub.1, the number of accessible diffraction lobes contained within the circle of radius NW/.lambda..sub.1 increases, thereby also increasing the diffusion. A one-dimensional reflection-phase grating with horizontal wells will scatter in directions represented by a vertical line in the reciprocal lattice reflection phase grating (with n=0, .+-.1, .+-.2, etc. and m=0) and diffraction from a one-dimensional reflection-phase grating with vertical wells will occur along a horizontal line (with m=0, .+-.1, .+-.2, etc. and n=0). A coordinate on the reciprocal lattice reflection phase grating plot is a direction. These scattering directions can be seen in the three-dimensional "banana" plot of FIG. 8, where the nine diffraction orders occurring within a circle of radius NW/.lambda..sub.2 are plotted, from a diagonal view perspective. A conventional polar pattern for a one-dimensional reflection-phase grating with vertical wells at .lambda..sub.2 is obtained from a planar slice through lobes 0, 2 and 6 in FIG. 8 and would contain orders with m=0 and .+-.1. The breadth of the scattering lobes is proportional to the number of periods contained in the reflection-phase grating.